Equilibrium in a Production Economy
Transcript of Equilibrium in a Production Economy
Equilibrium in a Production Economy
Prof. Eric Sims
University of Notre Dame
Fall 2013
Sims (ND) Equilibrium in a Production Economy Fall 2013 1 / 22
Production Economy
Last time: studied equilibrium in an endowment economy
Now: study equilibrium in an economy with production
Will produce operational model that can be used to compare to theactual behavior of the economy in the short run
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Equilibrium
Definition still the same: set of prices and quantities consistent with(i) agents optimizing, taking prices as given, and (ii) markets clearing
Agents: household, firm, government
Large number of each kind of agent, but identical: price-takingbehavior, can study representative agent problem
Time lasts for two periods: present, t, and future, t + 1
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Firm
Produce output using Yt = AtF (Kt ,Nt)
Take real wage, wt , as given
Time subscript on At : allow it to change period-to-period
Different than Solow model, assume that firms own capital stock andmake capital accumulation (investment) decisions
Would get same results if household owned capital stock as in SolowModel
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Capital Accumulation
Same equation as before, with one twist:
Kt+1 = qIt + (1− δ)Kt
q: investment-specific productivity.
Measure of how good we are at transforming investment into capitalOne way to think about financial system healthAssume it is the same in t and t + 1, differently than At
Terminal condition: Kt+2 = 0 ⇒ It+1 = − (1−δ)Kt+1
q . Intuition.
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Profits and Firm Value
Profit: Πt = Yt − wtNt − It
Firm value: present value of profit/dividend:
Vt = Πt +1
1 + rtΠt+1
Firm: picks Nt , Nt+1, and Kt+1 to maximize Vt
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Firm First Order Conditions
Optimality conditions:
wt = AtFN(Kt ,Nt)
wt+1 = At+1FN(Kt+1,Nt+1)
1 =1
1 + rt(qAt+1FK (Kt+1,Nt+1) + (1− δ))
Intuition: marginal benefit = marginal cost
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Labor Demand
First two first order conditions imply labor demand curves
Labor demand is “static”: depends only on current period stuff
Decreasing in the real wage
Labor demand shifts out if At goes up
Labor demand would shift in if Kt were destroyed (natural disaster)
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Investment Demand
The last first order condition implicitly defines an investment demandcurve
Investment a decreasing function of rt
Curve shifts out if At+1 or q go up
Curve also shifts out if Kt goes down exogenously (natural disaster)
Investment fundamentally forward-looking
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Household
Problem basically the same, but now household chooses amount oflabor/leisure
Normalize total endowment of time to 1 each period
Leisure is 1−Nt , where Nt is hours worked
Household gets utility from leisure via v(1−Nt), withv ′(1−Nt) > 0 and v ′′(1−Nt) ≤ 0
Lifetime utility:
U = u(Ct) + v(1−Nt) + β (u(Ct+1) + v(1−Nt+1))
Sims (ND) Equilibrium in a Production Economy Fall 2013 10 / 22
Budget Constraints
Basically look same, but have to account for endogenous income now
Household income comes from wages, dividend/profit from firm, andpays taxes to government
Ct + St = wtNt − Tt + Πt
Ct+1 = wt+1Nt+1 − Tt+1 + Πt+1 + (1 + rt)St
Combine into one:
Ct +Ct+1
1 + rt= wtNt − Tt + Πt +
wt+1Nt+1 − Tt+1 + Πt+1
1 + rt
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Household First Order Conditions
Household chooses Ct , Ct+1, Nt , and Nt+1 to maximize lifetimeutility. Optimality conditions:
u′(Ct) = β(1 + rt)u′(Ct+1)
v ′(1−Nt) = u′(Ct)wt
v ′(1−Nt+1) = u′(Ct+1)wt+1
Consumption Euler equation: same as it ever was
Two new conditions: implicitly define labor supply curves
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Labor Supply
Condition v ′(1−Nt) = u′(Ct)wt implicity defines labor supply curve
Can analyze in indifference curve-budget line diagram
Changes in wt : complicated effect because offsetting income andsubstitution effects
Assume that substitution effect dominates: Nt increasing in wt
Simple rational: MPC is less than 1, so Ct reacts less thanone-for-one to one period change in wt
Easy to see with log utility over consumption
Labor supply will shift with anything which affects Ct other than wt
To make life easy, assume that only thing that shifts Ns is rt : higherrt , N
s shifts out
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The Government
Same as before. Gt and Gt+1 chosen exogenously
Government’s intertemporal budget constraint:
Gt +Gt+1
1 + rt= Tt +
Tt+1
1 + rt
Ricardian Equivalence holds: household behaves as thoughgovernment balances budget every period
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Equilibrium Conditions
Labor demand: Nd = N(wt ,At ,Kt)
Labor supply: Ns = N(wt , rt)
Consumption: Ct = C (Yt − Gt ,Yt+1 − Gt+1, rt)
Investment: It = I (rt , q,At+1,Kt)
Production function: Yt = AtF (Kt ,Nt)
Market-clearing: Yt = Ct + It + Gt
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The Y s Curve
Set of (rt ,Yt) pairs consistent with production function where labormarket clears
Basic idea of derivation:
Start with an initial rt . Determines a position of Ns
Try a higher rt . Leads to labor supply shifting out. Higher Nt →higher Yt
Hence, Y s slopes up – higher rt effectively makes people want to workmore, and hence supply more output
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The Y d Curve
Set of (rt ,Yt) pairs consistent with agent optimization and Y dt = Yt ,
where Y dt = Ct + It + Gt
Basic idea of derivation:
Use the expenditure line - 45 degree line diagram. Start with an rt ,determines position of expenditure lineIncrease rt . Causes expenditure line to shift down – both because of Ct
and It . Intersects 45 degree line at lower pointHence, Y d
t slopes down
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General Equilibrium
General equilibrium requires that all markets clear
Effectively two markets here: labor (Ns = Nd) and goods (Y d = Y )
Labor market-clearing: on Y s curve
Goods market-clearing: on Y d curve
General equilibrium: on both curves
Real interest rate, rt , links “goods market” (Y d − Y s) with “labormarket” (Nd −Ns)
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Equilibrium: Graphically
Wt
Wt0
Yt
Nt0 Nt
Nt Yt
Yt0
45°
Yt
Yt
Yt
rt0
rt
Ytd
Yd(r)
Ys
Yd
Yt=Yt Yt=AF(Kt,Nt)
Nd
Ns
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Curve Shifts
Effectively five exogenous variables: At , At+1, q, Gt , and Gt+1
What shifts what:Labor demand: shifts if either At increases or Kt declines suddenly(natural disaster)
Note caveats about effects of At , q, Gt and their indirect effects onYt+1!
Goods demand: shifts if At+1, q, Gt , or Gt+1 change
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Analyzing Effects of Changes in Exogenous Variables
Follow cookbook approach:
Start in labor market. See if Nt would change for a given rt . Tells youif Y s curve shiftsFigure out if Y d curve shiftsCombine to find new equilibrium (rt ,Yt)Figure out what happens to components of Yt
Work back to labor market to make quantities line up
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Qualitative Effects
Variable: ↑ At ↑ At+1 ↑ q ↑ Gt ↑ Gt+1
Output + + + + -
Hours ? + + + -
Consumption + ? ? - -
Investment + ? + - +
Real interest rate - + + + -
Real wage + - - - +
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